Are there, moreover, only finitely many deformation types for $|e(X)| = (-1)^{n}e(X)$ bounded?

For $n = 2$, we have of course the famous Bogomolov-Miyaoka-Yau bound $c_1^2 \leq 3c_2$.

Remark. One may also include the convention $c_0 := 1$, for which the answer is positive since $(-1)^nc_n > 0$, in order to cover the prototypical observation that the Euler number of a hyperbolic surface is negative. In view of this example, I would also like to extend the original question to the realm of log manifolds and hyperbolicity.

I ask about bounding the degree-$n$ combinations of the $c_i$ in terms of $c_n$. Now, $c_i$ has degree $i$. Since the only degree-$n$ combination involving $c_n$ is $c_n$ itself, I removed $i=n$ from the product.
–
Vesselin DimitrovFeb 9 '13 at 22:40

Proposition 3.13 in this paper of Catanese-Schneider dx.doi.org/10.1007/BF01444736 gives universal bounds for the Chern numbers (assuming $K_X$ ample like you want) in terms of $(−1)^n c^n_1=K^n_X$. Does this help? When $n=3$, you also have the Yau inequality which bounds −$c^3_1\leq(8/3)(−c_1)c_2$, but then?
–
YangMillsFeb 12 '13 at 1:46